Question

Group theory Q. let f: A->B and g: B-> C. prove that if g o f is surjective, then g is surjective

Answer 1

This doesn't contain anything specific to group theory... just looks like basic knowledge of functions :-p

Answer 2

its definitely a group theory class...i am not sure how to show subjective
when everything is arbitrary.

Answer 3

Let \(f:A\to B,g:B\to C\). Given \(g\circ f\) is surjective, there exists some \(a\in A\) such that \(g(f(a))=c\) for all \(c\in C\). Since \(f[A]\in B\), it follows that there exists some \(b\in B\) (namely \(f(a)\) for some \(a\)) such that \(g(b)=c\) for all \(c\in C\). Therefore \(g\) is surjective.

Answer 4

I'm not doubting it's a group theory class... I'm just saying it's not a group-specific question :-p

Answer 5

its my first upper level math...with no previous proof experience...so the abstractness plus proofing is a challenge

Answer 6

Yeah, I can understand that... do you understand my approach though?

Answer 7

oops, I meant \(f[A]\subseteq B\) by the way! this just means that for all \(a\in A\) we have \(f(a)\in B\)

Answer 8

I'm working through it now...thank you. i get the general idea....but putting into notation and structure as you did will take time on my end :) thank you!!!